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Mathematics Stack Exchange Q A for people studying math at any level and professionals in related fields
(Un-)Countable union of open sets - Mathematics Stack Exchange A remark: regardless of whether it is true that an infinite union or intersection of open sets is open, when you have a property that holds for every finite collection of sets (in this case, the union or intersection of any finite collection of open sets is open) the validity of the property for an infinite collection doesn't follow from that In other words, induction helps you prove a
The sequence of integers - Mathematics Stack Exchange Prove that the sequence $\\{1, 11, 111, 1111, \\ldots\\}$ will contain two numbers whose difference is a multiple of $2017$ I have been computing some of the immediate multiples of $2017$ to see how
Prove that the sequence (1+1 n)^n is convergent [duplicate] It is hard to avoid "the concept of calculus" since limits and convergent sequences are a part of that concept On the other hand, it would help to specify what tools you're happy with using, since this result is used in developing some of them (For example, if you define ex = limn → ∞(1 + x n)n, then clearly we should not be using ex in the process of showing that this limit exists )
$\\operatorname{Aut}(\\mathbb Z_n)$ is isomorphic to $U_n$. It might be using ring theory in a non-essential way, but it is conceptually simpler because the endomorphisms are easier to describe than the automorphisms, and since the invertible elements of Zn Z n are by definition Un U n, we obtain the result without having to understand what Un U n actually looks like
Homotopy groups U(N) and SU(N): $\\pi_m(U(N))=\\pi_m(SU(N))$ As for your request regarding a table of the homotopy groups of SU(N) S U (N), the groups πm(SU(N)) π m (S U (N)) for 1 ≤ m ≤ 15 1 ≤ m ≤ 15 and 1 ≤ N ≤ 8 1 ≤ N ≤ 8 are given in appendix A, section 6, part VII of the Encyclopedic Dictionary of Mathematics This doesn't quite cover all the cases you asked for in the comment below, but the missing ones follow from complex Bott